Elastohydrodynamic aspects on
the tyre-pavement contact at
aquaplaning
Peter Andrén
Alexei Jolkin
VTI r
appor
t 483A • 2003
VTI rapport 483A · 2003
Omslagsbild: Peter Andrén, VTI
Elastohydrodynamic aspects on the tyre-pavement
contact at aquaplaning
Peter Andrén
Alexei Jolkin
ISSN: 0347–6030 Language: English No. of pages: 47
Abstract (background, aims, methods, results) max 200 words:
The objective of the work presented in this report has been to develop a numerical method for the inves-tigation of water-lubricated soft elastohydrodynamic (EHD) conjunctions as it relates to the problem of car tyre aquaplaning.
Whereas the problem of aquaplaning on very thick water films is very well-studied, the aspects of a tyre sliding on thin water layer is much less investigated. Considering fluid flow in very thin layers, one can find that the relative importance of viscosity of the fluid increases over the mass inertia effects. Therefore it is suggested in the present investigation to distinct between two different regimes of aquaplaning and refer them as dynamic aquaplaning and viscous aquaplaning. The subject of the present investigation is viscous aquaplaning on thin fluid layer.
Considering flow in a thin fluid layer allows a number assumptions to be made, which brings this problem close to the problem of fluid lubrication of machine elements. This makes it possible to apply advanced numerical methods originally developed for the lubrication theory to the aquaplaning of a pneumatic car tyre on a thin layer of water.
Numerical results were produced for 20, 40, 60, 120, and 200 km/h. No fluid films were detected separating the surfaces in this range of velocities.
Title:
Elastohydrodynamic aspects on the tyre-pavement contact at aquaplaning Author:
Peter Andr´en and Alexei Jolkin
Sponsor: VINNOVA Publisher:
SE-581 95 Link¨oping, Sweden Project:
Elastohydrodynamic aspects on the tyre-pavement contact at aquaplaning Published:
2003
Project code: 80 394
Publication: VTI report 483A
ISSN: 0347–6030 Spr˚ak: Engelska Antal sidor: 47
Referat (bakgrund, syfte, metod, resultat) max 200 ord:
M˚alet med arbetet som presenteras i denna rapport har varit att utveckla en numerisk metod f¨or att unders¨oka hur kunskapen om vattensm¨orjd mjuk elastohydrodynamisk (EHD) kontakt kan anv¨andas f¨or problemst¨allningen kring ett bild¨acks vattenplaning.
Medan problemen med vattenplaning p˚a ett tjockt lager vatten ¨ar v¨alstuderat, kvarst˚ar mycket att g¨ora n¨ar det g¨aller vattenplaning p˚a ett tunt lager vatten. F¨or tunna v¨atskefilmer g¨aller att den relativa effek-ten av viskositeeffek-ten ¨okar ¨over tr¨oghetseffekterna. I denna rapport kommer det att skiljas mellan dessa tv˚a typer av vattenplaning, vilka kommer att kallas dynamisk vattenplaning och visk¨os vattenplaning.
¨
Amnet f¨or denna unders¨okning ¨ar visk¨os vattenplaning p˚a ett tunt lager vatten.
Vissa antaganden g¨or att detta problem liknar problemet f¨or sm¨orjning av maskinelement, vilket g¨or det m¨ojligt att anv¨anda de avancerade numeriska metoder som ursprungligen utvecklades f¨or sm¨orjningsteori till visk¨os vattenplaning.
Numeriska resultat producerades f¨or 20, 40, 60, 120, och 200 km/h. Ingen separerande v¨atskefilm uppt¨acktes f¨or dessa hastigheter.
Titel:
Elastohydrodynamiska aspekter p˚a kontakten mellan d¨ack och v¨agbana vid vattenplaning F¨orfattare:
Peter Andr´en och Alexei Jolkin
Uppdragsgivare: VINNOVA Utgivare:
581 95 Link¨oping Projektnamn:Elastohydrodynamiska aspekter p˚a kontakten
mellan d¨ack och v¨agbana vid vattenplaning Utgivnings˚ar:
2003
Projektnummer: 80 394
Publikation: VTI rapport 483A
Preface
This project was initiated four years ago by Henrik ˚Astr¨om, at that time working
at VTI. The project was funded by the Swedish Agency for Innovation Systems (VINNOVA).
Henrik left VTI about two years ago, and from this point I have been the project manager. Though, even after leaving VTI Henrik has been heavily involved in the project, and his literature survey — which serves as an excellent background to
the material presented in the present report — was published seperately1. Alexei
Jolkin has made an incredible work on the elastohydrodynamic solver — the very key ingredient of the project. I thank you both for your excellent work, and good company.
Many more people have been involved in the project and I thank Arne Land, H˚akan V˚angenbrant, Leif Lantto, Lars-Gunnar Stadler, Roland Jacobsson, Roumald Banek and Sven- ˚Ake Lind´en for their participation.
Link¨oping, October 2003
Peter Andr´en
1“Elastohydrodynamiska aspekter p˚a vattenplaning — En litteraturstudie”. See [12] in reference
Contents
List of Figures 5
Summary 7
Sammanfattning 11
1 Introduction 15
2 Modelling of the Car Tyre 17
2.1 Meshing the Tyre 17
2.2 Finite Element Method (ABAQUS) 18
2.3 Results from ABAQUS 19
2.4 The Green Function Approach 22
3 Theory of Elastohydrodynamics 23
3.1 Governing equations 23
3.1.1 The Reynolds Equation 23
3.1.2 Film thickness equation 24
3.1.3 Force Balance equation 24
3.2 Discrete Equations 24
3.3 Numerical approach 26
3.3.1 Gauss-Seidel Line Relaxation 27
3.4 Jacobi Distributive Line Relaxation 29
4 Elastic Models of a Car Tyre 32
4.1 Solid Isotropic Car Tyre Model 32
4.2 Pneumatic Car Tyre Model 33
4.3 Additional Remarks on Car Tyre Models 35
5 Results 36
5.1 General remarks on the viscous EHD pressures and contact
shapes 36
5.2 Running conditions for the numerical calculations 37
5.3 Analysis of simulation results 40
5.4 Prediction of the EHD fluid layer thickness by the classical
the-ory of lubrication 40
6 Discussion 42
7 Conclusions 45
8 Future Investigations 46
Bibliography 47
List of Figures
2.1 Tyre outlines for the P.I.A.R.C. smooth test tyre. 17
2.2 Nodes in the complete tyre section meshes. 18
2.3 Element sets in the complete tyre section meshes. 19 2.4 Inflated model of the cross-section of the tyre. 19
2.5 Examples of the complete finite element model. 20
2.6 Comparison between experiment and simulation of P.I.A.R.C. test tyre. 20 2.7 Comparison of contact areas of experiment and simulation. 21 2.8 Examples of deformation and stress disribution from unit loads. 22
3.1 Part of a car tyre in contact with the road. 25
3.2 EHL conjunction. Choice of relaxation scheme for different grid points. 27 3.3 Gauss-Seidel Line relaxation, and Jacobi distributive line relaxation. 28 4.1 Path of a car tyre in contact with the plane rigid surface. 33 4.2 Shape of the car tyre in contact with road, and corresponding pressure
distribution. 34
4.3 Illustration on Green function of the elastic model of the pre-loaded pneumatic car tyre. An elastic response of the car tyre surface to the point unit load. Deflections of all grid nodes from the initial
pre-loaded state are shown on a coarse grid. 34
5.1 Illustration on static and dynamic, 1.66 × 103 m/s elastic contacts of solid isotropic elastic car tyre with flat rigid surface. 38 5.2 Illustration on static and elastohydrodynamic at 1.66 × 103 m/s
con-tacts of pneumatic car tyre with flat solid surface. 39 5.3 Predicted fluid film thickness by Soft-EHL theory. 41
Elastohydrodynamic aspects on the tyre-pavement contact at aquaplaning by Peter Andr´en and Alexei Jolkin
Swedish National Road and Transport Research Institute (VTI) SE-581 95
Summary
The objective of the work presented in this report has been to develop a numeri-cal method for the investigation of water-lubricated soft elastohydrodynamic (EHD) conjunctions as it relates to the problem of car tyre aquaplaning. Aquaplaning oc-curs when a vehicle rides on the water and completely loses contact with the road, putting drivers in immediate danger of sliding of the road. The factors that most contribute to aquaplaning are vehicle speed, vehicle weight, water depth, tyre size, tyre treads depth, tyre tread pattern, and water composition.
The phenomenon when the tyre loses contact with the road while driving at high speed through a deep puddle, i.e. through large amounts of water, is well-known and generally referred to as aquaplaning. This phenomenon has been a subject of numerous investigations performed by the world’s leading tyre manufacturers and other researchers. The problem is mainly of hydrodynamic nature and the stud-ies concern mainly flow of very large volumes of water through the contact zone between the tyre and the road.
However, the fact remains that even for relatively thin layers of water present on the road surface, the traction in the tyre to road contact dramatically decreases. This in many cases may lead to very similar dangerous consequence as for the case with deep water layers, which is total loss of control over the vehicle.
Whereas the problem of aquaplaning on very thick water films is very well-studied, the aspects of a tyre sliding on a thin water layer is much less investigated. Considering fluid flow in very thin layers, one can find that the relative importance of viscosity of the fluid increases over the mass inertia effects. Therefore it is sug-gested in the present investigation to distinct between two different regimes of aqua-planing and refer them as dynamic aquaaqua-planing and viscous aquaaqua-planing.
The subject of the present investigation is viscous aquaplaning on a thin fluid layer, where water acts as a lubricant on the rubber. Due to the complexity of the entire problem, a number of assumptions are used in the present initial study,
• A single perfectly smooth tyre
• A perfectly smooth and infinitely stiff road surface • A very thin water film
• No mass inertia effects • Pure water lubrication
These assumptions brings this problem close to the problem of fluid lubrication of machine elements, which makes it possible to apply advanced numerical methods originally developed for the lubrication theory to the viscous aquaplaning of a pneu-matic car tyre.
The tyre used in the calculation is a P.I.A.R.C. test tyre. An outline of the smooth P.I.A.R.C. tyre was taken from a construction drawing. Due to the non-availability of precise information on this tyre the theoretical model is that of a generic radial ply tyre with the correct dimensions.
To generate the mesh needed for the finite element computations the MetaPost program was used to generate a spline interpolated outline. A full mesh was then generated in MATLAB. The node coordinates and element information was writ-ten to text files, used by the finite element program ABAQUS. The MetaPost to MATLAB to ABAQUS chain was very useful in the evaluation phase of the cod-ing. The low level of human interaction and high level of feedback made the de-bugging simple, and only needed in the beginning of the project. Once the code worked, many different mesh sizes, loading conditions, etc. could be tested in a very straight-forward manner.
The full finite element tyre model was validated with a controlled loading ex-periment on a real test tyre and on the simulated tyre. The agreement between the experiment and the simulation was very satisfactory, with only minor discrepancies at higher loads. Both the contact area and the shape of the contact zone is almost perfect. The elastic behaviour of the model is synthesized in a “flexibility matrix”. In order to make the complete matrix a unit load is applied, one at the time, to all nodes in the contact region. Assuming local linearity, the total deformation of any pressure distribution can be calculated by superimposing the deformations from the unit loads.
The mathematical elastohydrodynamic model allows the water film shape and pressure in the contact between the tyre and the road to be determined and analysed. This model comprises the Reynolds Equation, the film thickness equation and the force balance equation. The equations are discretized on a rectangular uniform grid and solved numerically using an approximate relaxation method. Understanding how to solve the discretized Reynolds Equation forms the key to understanding how to construct an efficient solver for the complete elastohydrodynamic lubrication problem. No single relaxation process is suited to efficiently solve the problem for all situations, so a combination of the Jacobi and Gauss-Seidel relaxation methods is used.
Two different elastic material models of the car tyre were considered, with an increasing degree of complexity. First, a solid car tyre modelled as a homogeneous, isotropic and perfectly elastic body. Second, the more realistic model is the pneu-matic car tyre modelled in finite elements described above.
Numerical results were produced for 20, 40, 60, 120, and 200 km/h. No fluid films were detected separating the surfaces in the range of chosen velocities. In the range of velocities from 20 to 200 km/h the contact shapes and pressures were al-most exactly identical with dry static contact. As a matter of fact, the velocity has to be around 1.6×103m/s when the first full film regime is achieved. This conclusion is valid for both elastic models. Prediction of the EHD fluid layer thickness by the classical theory of lubrication also shows, even though it somewhat overestimates the film thickness for the isoviscous and incompressible case, that the effects of viscous flow are very small.
The physics of water films is completely different from the ones build by lubri-cating oils. Due to its dense molecular structure, water undergoes no or very little compression even under extreme pressures. Pure water is a very bad lubricant and
is indeed very rarely used in highly loaded concentrated conjunctions. This is be-lieved to be the major reason why no separating fluid films were detected in the pure viscous water aquaplaning. Another mechanism that can contribute to lose of grip already on a slightly wet road can be described as boundary lubrication. Boundary lubrication is a field of knowledge that combines tribology, chemistry and material science.
Elastohydrodynamiska aspekter p˚a kontakten mellan d¨ack och v¨agbana vid vattenplaning
av Peter Andr´en och Alexei Jolkin
Statens v¨ag- och transportforskningsinstitut (VTI) SE-581 95
Sammanfattning
M˚alet med arbetet som presenteras i denna rapport har varit att utveckla en nu-merisk metod f¨or att unders¨oka hur kunskapen om vattensm¨orjd mjuk elastohydro-dynamisk (EHD) kontakt kan anv¨andas f¨or problemst¨allningen kring ett bild¨acks vattenplaning. Vattenplaning intr¨affar n¨ar ett fordon planar p˚a vattnet och helt och h˚allet f¨orlorar v¨aggreppet, vilket uts¨atter f¨oraren f¨or en omedelbar risk att k¨ora (eller snarare glida) av v¨agen. De f¨orh˚allanden som fr¨amst bidrar till vattenplan-ing ¨ar fordonets hastighet och vikt, vattendjup, d¨ackstorlek, d¨ackets m¨onsterdjup och m¨onsterutformning, samt vattnets sammans¨attning.
Med vattenplaning avses oftast det tillst˚and som intr¨ader n¨ar ett d¨ack f¨orlorar kontakten med v¨agbanan d˚a det k¨or genom en tillr¨ackligt stor m¨angd vatten. Detta fenomen har varit f¨orem˚al f¨or en stor m¨angd unders¨okningar gjorda av v¨arldens ledande d¨acktillverkare och andra forskare. Problemet utg¨ors till st¨orsta del av hy-drodynamiska effekter, d.v.s. fl¨odet av stora m¨angder vatten genom kontaktzonen mellan d¨acket och v¨agbanan.
Dock kvarst˚ar faktum att v¨aggreppet reduceras rej¨alt ¨aven f¨or ett relativt tunt vattenlager p˚a en sl¨at v¨agbana. Detta kan leda till situationer som liknar de vid st¨orre vattendjup, allts˚a f¨orlorad kontroll av fordonet.
Medan problemen med vattenplaning p˚a ett tjockt lager vatten ¨ar v¨alstuderat, kvarst˚ar mycket att g¨ora n¨ar det g¨aller vattenplaning p˚a ett tunt lager vatten. F¨or tunna v¨atskefilmer g¨aller att den relativa effekten av viskositeten ¨okar ¨over tr¨oghets-effekter. I denna rapport kommer det att skiljas mellan dessa tv˚a typer av vatten-planing, vilka kommer att kallas dynamisk vattenplaning och visk¨os vattenplaning.
¨
Amnet f¨or denna unders¨okning ¨ar allts˚a visk¨os vattenplaning p˚a ett tunt lager vatten, d¨ar vattnet agerar som ett sm¨orjmedel i kontakten mellan d¨ack och v¨agbana. P˚a grund av komplexiteten f¨or hela problemet har n˚agra antaganden gjorts f¨or denna inledande studie.
• Ett helt sl¨att d¨ack
• En helt sl¨at och o¨andligt styv v¨agbana • En v¨aldigt tunn vattenfilm
• Inga tr¨oghetseffekter • Ren vattensm¨orjning
Dessa antaganden g¨or att detta problem liknar det f¨or sm¨orjning av maskinelement, vilket g¨or det m¨ojligt att anv¨anda de avancerade numeriska metoder som ursprung-ligen utvecklades f¨or sm¨orjningsteori till visk¨os vattenplaning.
D¨acket som anv¨ants i ber¨akningarna ¨ar ett P.I.A.R.C. testd¨ack. En kontur av d¨acket togs fr˚an en konstruktionsritning. P˚a grund av otillg¨angligheten av exakt information om d¨acket gjordes den teoretiska modellen som ett allm¨ant radiald¨ack med dimensionerna fr˚an P.I.A.R.C.-d¨acket.
F¨or att skapa elementindelningen som beh¨ovs f¨or ber¨akningarna med finita ele-mentmetoden anv¨andes programmet MetaPost f¨or att skapa en splineinterpolerad kontur av d¨acket. Den fullst¨andiga elementindelningen skapades sedan i MATLAB. Nodkoordinaterna och elementinformationen skrevs till textfiler, som anv¨andes av det finita elementprogrammet ABAQUS f¨or att skapa modellen. Programkedjan MetaPost till MATLAB till ABAQUS var mycket nyttig i utvecklingsfasen av ko-den. Den l˚aga niv˚an av m¨anskligt redigerande och h¨oga niv˚an av ˚aterkoppling har minimerat m¨ojligheterna f¨or fel och gjort debuggningen av programmen enkel, och dessutom endast n¨odv¨andig i b¨orjan av projektet. N¨ar koden v¨al fungerade kunde m˚anga olika elementindelningar, lastfall, etc. testas p˚a ett mycket enkelt och bekv¨amt s¨att.
Den kompletta elementindelningen f¨or d¨acksmodellen validerades med ett last-kontrollerat experiment p˚a att riktigt testd¨ack och p˚a det simulerade d¨acket. ¨ Over-ensst¨ammelsen mellan experiment och simulering var v¨aldigt f¨ortroendeingivande, med endast mindre avvikelser f¨or h¨ogre laster. B˚ade kontaktarean och formen f¨or kontaktomr˚adet ¨ar n¨astan perfekt.
De elastiska egenskaperna hos modellen samlas i en ”flexibilitetsmatris”. F¨or att skapa en hel matris p˚af¨ors en enhetslast, en i taget, p˚a alla noder i kontaktomr˚adet. Med antagande om lokal linj¨aritet, kan deformationen fr˚an vilket tryck som helst ber¨aknas genom att superpositionera deformationerna fr˚an de enskilda enhetsfallen. Den matematiska elastohydrodynamiska modellen ber¨aknar formen p˚a vattenfil-men och trycket i kontakten mellan d¨ack och v¨agbana. Modellen best˚ar av Reynolds Ekvation, filmtjockleksekvationen och kraftj¨amviktsekvationen. Ekvationerna disk-retiseras p˚a ett rektangul¨art enhetligt indelat rutn¨at, och l¨oses med en approximativ relaxeringsmetod. F¨orst˚aelse f¨or l¨osningen av den diskretiserade Reynolds Ekvation ¨ar nyckeln till f¨orst˚aelse f¨or att konstruera en effektiv l¨osningmetod f¨or det sam-lade elastohydrodynamiska sm¨orjningsproblemet. Ingen enskild relaxeringsmetod ¨ar l¨ampad att effektivt l¨osa detta problem f¨or alla situationer, s˚a en kombination av Jacobis och Gauss-Seidels relaxeringsmetod anv¨ands.
Tv˚a olika elastiska modeller av bild¨acket anv¨andes i ber¨akningarna. F¨orst, ett solitt bild¨ack vilket modelleras som en homogen, isotropisk och helt elastisk kropp. Den andra modellen ¨ar det betydligt mer realistiska luftfyllda d¨ack som gjort med finita elementmodellen enligt beskrivningen ovan.
Numeriska resultat producerades f¨or 20, 40, 60, 120, och 200 km/h. Ingen se-parerande v¨atskefilm uppt¨acktes f¨or dessa valda hastigheter. F¨or intervallet 20 till 200 km/h var kontaktformen och -trycket i det n¨armaste identiskt med vad som f˚as vid torr statisk belastning. Faktiskt m˚aste hastigheten vara runt 1,6×103m/s innan den f¨orsta separerande v¨atskefilmen erh˚alls. Denna slutsats g¨aller f¨or b˚ada elastiska modellerna. Uppskattningar av filmtjockleken med klassisk sm¨orjningsteori visar ocks˚a, trots att filmtjockleken ¨overskattas n˚agot f¨or det isovisk¨osa och inkompress-ibla fallet, att effekterna av det visk¨osa fl¨odet ¨ar mycket sm˚a.
Fysiken f¨or vattenfilmer skiljer sig radikalt fr˚an de som skapas av sm¨orjoljor. P˚a grund av dess t¨ata molekylstruktur komprimeras vatten i princip ingenting ¨aven un-der extrema tryck. Rent vatten ¨ar ett v¨aldigt d˚aligt sm¨orjmedel och anv¨ands mycket
s¨allan i kontakter med h¨og last. Detta antas vara huvudanledningen till att ingen separerande vattenfilm uppt¨acktes f¨or den rent visk¨osa vattenplaningen. En annan mekanism som kan leda till f¨orlorat v¨aggrepp ¨aven p˚a en endast n˚agot v˚at v¨ag ¨ar gr¨ansskiktssm¨orjning. Gr¨ansskiktssm¨orjning ¨ar en vetenskapsgren som innefattar tribologi, kemi och materiall¨ara. Det ligger dock utanf¨or detta projekts ramar att unders¨oka denna effekt.
1
Introduction
Aquaplaning (called hydroplaning in North America) occurs when water on the roadway accumulates in front of the tyres of a vehicle faster that the weight of the vehicle can push it out of the way. The water pressure can cause the car to rise up and slide on top of a thin layer of water between the tyres and the road. While aquaplaning the vehicle rides on top of the water and can completely lose contact with the road, putting road-users in immediate danger of sliding out of the lane.
The following factors contribute to aquaplaning
• Vehicle speed — as speed increases, wet traction is considerably reduced.
Aquaplaning can result in a complete loss of traction and vehicle control.
• Vehicle weight — the lighter the vehicle, the more likely it is to aquaplane. • Road surface type — non-grooved asphalt is considerably more
aquaplane-prone than ribbed or grooved concrete surfaces.
• Water depth — the deeper the water, the sooner you will lose traction,
al-though thin water layers can cause a loss of traction, even at low speeds.
• Tyre size — the size and shape of the contact patch has a direct influence on
the probability to aquaplane. The wider the contact patch is relative to its length, the higher the speed required supporting aquaplaning.
• Tyre treads depth — as the tyres become worn, their ability to resist
aqua-planing is reduced.
• Tyre tread pattern — certain tread patterns channel water more efficiently,
reducing the risk of aquaplaning.
• Water composition — oil, temperature, dirt, and salt can change its properties
and density.
Let’s examine what happens to a tyre in the midst of a aquaplane. When entering a puddle, the surface of the tyre must move the water out of the way in order for the tyre to stay in contact with the pavement. The tyre pushes some of the water to the sides, and forces the remaining water through the tyre treads. On a smooth polished road in moderate rain at 90 km/h, each tyre has to displace about four litres of water every second from beneath a contact patch no bigger than a palm of a hand. Each gripping element of the tread is on the ground for only 1/150th of a second, during this time it must displace the bulk of the water, press through the remaining thin film, and then begin to grip the road surface.
With other words, water acts as a lubricant on rubber. This fact brings the car tyre aquaplaning problem close to the lubrication problem of machine elements. The major difference of aquaplaning research is to increase friction and eliminate the water lubricating layer in order to improve traffic security, whereas the lubrica-tion of machine components has its goals in reducing friclubrica-tion and wear by building a thick and stable lubricant film. The mathematical model is, however, very similar in both situations. The surfaces in the contact are deformed elastically and separated by a layer of lubricant, either oil or water. The separating film has a very complex shape formed by a lubricant (water) flow in a gap between the surfaces.
Due to the complexity of the entire problem, a number of assumptions are used in the present initial study.
• A system consisting of a single tyre loaded against a perfectly smooth road
surface is considered
• A perfectly smooth tyre is assumed. Although smooth tyres give better grip
on dry roads than treaded tyres, they are unsafe in rain. However, they are extensively used in different kind of research.
• A very thin water film between the tyre and the road is assumed, an
assump-tion of the so-called viscous aquaplaning. No inertia mass effects are consid-ered.
• Pure water lubrication is assumed
This model makes it possible to predict the contact pressure and the thickness of the separating water film under different circumstances, such as variable speed and load, sliding, etc.
2
Modelling of the Car Tyre
Automobile tyres are big business and lots of money and resources are spent on research. The large manufacturers most certainly have very sophisticated in-house models of their own tyres, but these are regarded as trade secrets and results very seldom published. Due to the structural complexity of a car tyre (hyperelasticity, viscoelasticity, large deformations, contact, steel reinforcement, etc.) it serves as a good bench-mark on finite element code. Some finite element modelling (FEM) im-plementations have been presented in, e.g., “Tire Science & Technology”. However, no detailed literature review of the tyre modelling literature has been made for this report. A good starting point for the interested is the following papers [3,5,8,13,4] where especially [5] contains references to early implementations.
To develop new finite element modelling code would be beyond the scope of
this project. Instead, the commercial software ABAQUS [1] has been used. The
theory or implementation of the FEM code will not be explained here. The theory is thoroughly described in the ABAQUS Theory Manual.
2.1
Meshing the Tyre
An outline of the smooth P.I.A.R.C. test tyre was taken from a construction drawing. (Exact specifications on the test tyre were hard to find. Number, positions and orientation of plies and material parameters were not available. The theoretical model is that of a generic radial ply tyre with the dimensions of the P.I.A.R.C. tyre.) Seven points along one half of the outline were carefully traced, and the results compared with an actual test tyre. The paths between these points were interpolated
with splines using the MetaPost program [7]. MetaPost was especially well-suited
as splines easily can be created with different “tension”. This made it possible to get a near perfect match with only the seven fixed points. Another advantage with MetaPost is that it is command line driven and can return the interesting results. In this way, it was possible to call the MetaPost program from within MATLAB
[11] with the dos() command, pipe the results through grep, and get the spline
interpolated points back. The result for three different mesh sizes can be seen in Figure2.1.
A full mesh was then generated in MATLAB. Depending on the number of in-terpolated points along the outline different numbers of layers of elements were created. The aim is, basically, to get the elements as square-shaped as possible.
(a) Coarse mesh. (b) Normal mesh. (c) Fine mesh.
Figure 2.1 Tyre outlines for the P.I.A.R.C. smooth test tyre. The seven fixed points
are marked with black, and the interpolated points with red.
(a) Coarse mesh. (b) Normal mesh. (c) Fine mesh.
Figure 2.2 Nodes in the complete tyre section meshes. Big black circles are rim
nodes, tread is green, belt is red, side is orange, and the rest are blue. The steel plies is shown in black dash-dotted lines.
The MATLAB program also divided the elements into four main element sets and the nodes into five node sets, as illustrated in Figure2.2 and2.3, respectively. The different element and node sets are used in the finite element analysis. In this model the sets have the following functions. The rim nodes are used to define, with bound-ary conditions, only to move in the “up-and-down” direction. The tread is needed to define an outside contact surface to the stiff foundation. An inflation pressure is applied to an inside surface defined on the side and belt elements. The belt and side set are also needed to host the embedded elements for the steel reinforcement. The belt and side sets both host the radial steel ply, and the belt set also hosts the two layers of belt ply. The node coordinates and element information was then written to text files, used by ABAQUS as input to construct the finite element model. The MetaPost to MATLAB to ABAQUS chain was very useful in the evaluation phase of the coding. The low level of human input of high level of feedback made the debugging simple, and only needed in the beginning of the project. Once the code worked, many different mesh sizes, loading conditions, etc. could be tested in a very straight-forward manner.
2.2
Finite Element Method (ABAQUS)
As said before, the tyre model used was constructed in the finite element modelling
frame-work using the commercial ABAQUS software [1]. Each element, or rather
element set, is defined as a certain element type.1 For the rubber parts a 4-node
bilinear, hybrid with constant pressure element is used (as a CGAX4H in ABAQUS language). This is defined as a both hyperelastic and viscoelastic material, with
material parameters taken from a published ABAQUS example [1]. Three steel ply
layers are defined. Two of the three can be seen as black bash-dotted lines in
Fig-ure2.2and2.3above. These two belt-layers are oriented at 20 degrees plus/minus
the longitudinal axis, creating a very stiff belt section. A third layer is placed at right angles (from rim to rim) on the inside of the side and belt element sets. The reinforcement elements are embedded in the “rubber” elements.
1In finite element modelling different element types incorporate different physical qualities
(a) Coarse. (b) Normal. (c) Fine.
Figure 2.3 Element sets in the complete tyre section meshes. Tread is green, belt is
red, side is orange, and the rest are blue. The steel reinforcement is shown in black dash-dotted lines in the belt set.
2.3
Results from ABAQUS
Below, in Figure2.4, is the deformed tyre after a pressure of 200 kPa. For increased visibility, the scale of the deformation is doubled, and the undeformed outline is superimposed (in black lines). We can see here that the deformation is largest at the sides. This is expected as the steel plies in the belt makes this part very stiff. Note also that the part where the rim normally would be is completely undeformed. The stress distribution for inflated tyre can also be seen below. (Actually, the steel reinforcement have much higher stresses, but these elements were removed in this picture in order to show the effect on the rubber parts.)
(a) Deformation (×2) (b) Stress distribution
Figure 2.4 Inflated model of the cross-section of the tyre. Colour codes as before.
A complete tyre is now made by revolving the tyre cross section. The mesh grid is more refined at the “bottom” part where the contact with the stiff road will occur. The contact between the tyre and the road is handled by defining an outer contact surface on the tyre and one on the road. Creating a complete tyre model is then just a matter of mirroring the half tyre, Figure2.5.
This full tyre model can now be validated. The finite element model was vali-dated by a loading comparison with an real P.I.A.R.C. test tyre. First the, to 200 kPa,
(a) Revolved half-tyre. (b) Stress distribution in the refined section.
Figure 2.5 Examples of the complete finite element model.
inflated test tyre was mounted on a loading rig. The tyre was loaded from about 1 kN to 6 kN, in steps of 1 kN, and the load measured both by the rig and by a scale placed under the wheel. The contact area was measured with a pressure sensitive graphite powder (see the left part of Figure2.7(a-l)). The corresponding simulation was per-formed in ABAQUS. The finite element model was inflated to 200 kPa. Then the tyre was loaded on a infinitely stiff surface with a load controlled algorithm to the same levels as in the experiment. The simulated contact pressure area was exported to a result file (see the right part of Figure2.7(a-l)), and compared with the results from the experiment. The simulated contact areas sometimes have a slight “angular” shape. This effect is caused by the fact that an element is either in contact or not. A finer mesh would, probably, have produced an even better agreement. A summary
is given in Figure 2.6. The agreement between the experiment and the simulation
is very satisfactory. In the elastohydrodynamic program only a small part at the
in-0 1000 2000 3000 4000 5000 6000 7000 0 0.5 1 1.5 2 2.5x 10 4 Load [N] Contakt area [mm 2 ] Laboratory experiment FEM−simulation
terval 2–3 kN will be of interest, so it’s safe to ignore the minor discrepancy at the higher loads. (a) 990 N (b) 1928 N (c) 2920 N (d) 3890 N (e) 4980 N (f) 6420 N (g) 5790 N (h) 4750 N (i) 3970 N (j) 2940 N (k) 1980 N (l) 790 N
Figure 2.7 Comparison of contact areas of experiment (left part of each sub-figure)
and simulation (right part of each sub-figure).
2.4
The Green Function Approach
For the model to be useful in the elastohydrodynamic program the elastic behaviour of the model must be synthesized in some way. Our approach has been to make a “flexibility matrix”. The flexibility matrix contains information on how much the tyre contact surface is deformed when a unit load is applied at some node. In order to make the complete matrix a unit load is applied, one at the time, to all nodes in the contact region. Assuming local linearity, the total deformation of any pressure distribution can be calculated by superimposing the deformations from the unit loads. Another way to obtain the same results would have been to call ABAQUS from within the elastohydrodynamic code. Apart from being more complicated, this would have been much to time consuming for practical use.
Four examples of deformations from the flexibility matrix creation is shown in
Figure 2.6. Because of the relative softness of the tyre most of the deformation
occur at the neighbouring nodes to the node where the unit load is applied. The magnitude of the deformation varies over the contact zone due to the initial stress from the loading.
(a) (b)
(c) (d)
3
Theory of Elastohydrodynamics
Fluid film lubrication occurs when two opposing surfaces are completely separated by a lubricant film that also carries the entire contact load. The subject of the present work is elastohydrodynamic lubrication (EHL) which is a mode of fluid film lubri-cation in which high contact pressure causes elastic deformation of the contacting surfaces of much higher order of magnitude than the thickness of the lubricant film separating them.
The earlier studies of elastohydrodynamic lubrication of elliptical conjunctions are applied to the different but interesting situation with a car tyre rolling on a wet surface. This situation exhibited by low-elastic modulus material leads to the so-called Soft-EHL. The procedure used in obtaining the Soft-EHL results, which is described by a number of authors (see references [6,2,10,14]), is applied in this study with minor modifications.
The objective of the work presented in this report is to develop a numerical method for the investigation of water-lubricated soft elastohydrodynamic conjunc-tions as it relates to the problem of car tyre aquaplaning.
3.1
Governing equations
The presented mathematical model allows the water film shape and pressure in the contact between the tyre and the road to be determined and analysed. This model comprises the Reynolds Equation, the film thickness equation and the force balance equation.
3.1.1 The Reynolds Equation
Knowing the velocities of the surfaces and their geometry, we can calculate the pressures in the water film with Reynolds Equation
∂ ∂x ρh3 η ∂p ∂x +∂y∂ ρh3 η ∂p ∂y = 6(U1+U2)∂(ρh)∂x (3.1)
where x and y are spatial coordinates along the direction of motion and in transverse
direction, respectively; p= p(x,y) is the pressure in the contact; h = h(x,y) is a
function describing the separation between two interacting bodies (the water film
thickness); ρ is the density of water; η is the viscosity of water; U1 and U2 are
surface velocities of the road and the tyre, respectively.
The following common assumptions are considered in the present investigation:
• Newtonian fluid • Iso-viscous fluid • Incompressible fluid • Steady-state • Isothermal • Smooth surfaces
• Fully flooded conditions
• Inertia effects neglected
We need to satisfy the following boundary conditions
at the inlet: p= 0
on each side of the contact: p= 0 (3.2)
at the cavitation boundary: p=∂p
∂x = 0
3.1.2 Film thickness equation
Irrespective of lubricant flow, rheological behaviour or thermal conditions, the film thickness in a two-dimensional EHL contact can be written as the sum of the unde-formed gap geometry, and the elastic deformation of the surfaces:
h(x,y) = h00+ g(x,y) + w(x,y) (3.3)
where h(x,y) is the water film thickness; g(x,y) is the shape of the undeformed tyre in the contact region, i.e. the gap between the undeformed tyre and the touching
plan; h00 is the film thickness at the origin of coordinates had the surfaces been
undistorted; and w(x,y) is the actual deformation of the tyre under action of external load and fluid flow.
The limited extent of tyre contact with the road makes it possible to consider only that part of the wheel which is close to the contact, see Figure3.1which shows the gap between the tyre and the road.
For two continuous bodies that are loaded against each other and separated by a thin water film, the total normal deflection w(x,y) from their undistorted shape can be written as the integral
w(x,y) = +∞ −∞ +∞ −∞ K(x,x ,y,y)p(x,y)dxdy (3.4)
where p(x,y) is the lubricant pressure acting over the contact between the surfaces. The integral kernel K(x,x,y,y) of a contact problem may take different forms depending on each elastic system studied. A car tyre is a rather complicated elastic system. A short description of the two different elastic models of the car tyre is given in Chapter4.
3.1.3 Force Balance equation
The force balance equation (3.5) imposes a global condition of equilibrium on the
film thickness equation, the calculated load must be equal to the value F, measured experimentally.
F =
Ω
p(x,y)dxdy (3.5)
By altering h00 the whole pressure distribution is raised or lowered until it satisfies
the force balance equation, the film thickness equation and the Reynolds Equation.
3.2
Discrete Equations
The equations are discretized on a rectangular uniform grid with mesh size hx and
discretisation of the Poiseuille terms and one-sided upstream second order discreti-sation of the Couette term gives the following discrete Reynolds Equation
Li, jp ≡(ei, j+ ei−1, j)pi−1, j− (ei−1, j+ 2ei, j+ ei+1, j)pi, j+ (ei, j+ ei+1, j)pi+1, j 2h2 x +(ei, j+ ei, j−1)pi, j−1− (ei, j−1+ 2ei, j+ ei, j+1)pi, j+ (ei, j+ ei, j+1)pi, j+1 2h2y =1.5ρi, jhi, j− 2ρi−1, jhi−1, j+ 0.5ρi−2, jhi−2, j hx (3.6) where ei, j= ρi, jh3i, j/6(U1+U2)ηi, j. Note, that this is a most general expression for
the discrete Reynolds Equation. In the assumptions above, ρi, j = ρ ≡ const and
ηi, j = η ≡ const.
Approximating the pressure profile by a polynomial of a certain degree, the in-tegral expression (3.4) can be integrated analytically (in very few cases) or numeri-cally. The integration results in a matrix ni× nj by ni × nj of influence coefficients
Dikjlfor each grid point(i, j). The influence coefficients show how the pressure in a discrete point(i, j) affects the deflections in all grid points. Thus, the equation (3.4) in the discrete form will appear as a multi-summation over all grid points
w(xi,yj) = wi, j≈ 2 πE nx
∑
k=1 ny∑
l=1 Dijklpkl. (3.7)The discrete film thickness equation is finally written as
Figure 3.1 Part of a car tyre in contact with the road.
hi, j = h00+ gi, j+ 2 πE nx
∑
k=1 ny∑
l=1 Dijklpkl (i = 1...nx, j = 1...ny). (3.8)The force balance equation is discretized in the same way as the film thickness equation, i.e. by approximating the pressures with a piecewise constant function on a uniform grid with mesh size hxand hy
F≈ hxhy nx
∑
i=1 ny∑
j=1 pi, j. (3.9)3.3
Numerical approach
In order to design a stable and efficient relaxation scheme it is necessary to under-stand the nature of the equations. Due to elastic effects the coefficient e in
equa-tion (3.6) varies many orders of magnitude over the domain Ω. For large values
of e the differential aspects as represented by the second order derivatives of the
pressure determine the behaviour. For small values of e the term ∂(ρh)/∂x
domi-nates the behaviour and, because the film thickness equation is given by an integral equation, the problem behaves as an integral problem. Understanding how to solve this problem for large and small values of e forms the key to understanding how to construct an efficient solver for the complete EHL problem.
Generally speaking, the system of non-linear equations (3.6, 3.8, 3.9) consists of a very large number of unknowns. Any direct method to solve this problem is of questionable applicability. This difficulty can be overcome by using the approx-imate relaxation method. There are many different iterative processes and it goes beyond the scope of this investigation to give a detailed overview. Two very basic processes referred to as Jacobi and Gauss-Seidel relaxation respectively are often used in practice.
In words the process can be described as follows. Given an approximation to the solution of the problem in each grid point, a new approximation is computed by scanning the grid points in a prescribed order, changing the value in each grid point such that the local equation in this point is satisfied. The process comes in two flavours. If the new values are computed using only old values in the surrounding grid points it is referred to as Jacobi relaxation (Simultaneous Displacement). If the changes made when relaxing previous grid points are taken into account when relaxing the next grid point it is referred to as Gauss-Seidel relaxation (Successive Displacement). For Jacobi relaxation the order in which the grid points are relaxed is irrelevant. For Gauss-Seidel relaxation it makes a difference. Often the grid points are scanned in order to increasing grid indices in which case one refers to it as Gauss-Seidel relaxation with lexicographic ordering.
However, neither of the relaxation processes is suited to efficiently solve the
problem for all values of e over the whole domainΩ. The Gauss-Seidel relaxation
is an excellent method for large values of e but unstable for small e. On the other hand the distributive line Jacobi relaxation is a good method for very small e but rapidly looses efficiency with increasing e. The objective is to obtain an efficient
solver. Let us introduce a parameterξ. A good choice to obtain an efficient solver
seems to be to use different relaxations for different grid points, as schematically
shown in Figure 3.2. This figure presents a typical pressure and film thickness
profiles in an elastohydrodynamically lubricated contact.
• On grid points where ξ/(hxhy) > ξlim the Gauss-Seidel line relaxation is used
• On grid points where ξ/(hxhy) ξlimthe Jacobi distributive line relaxation is used
Figure 3.2 EHL conjunction. Schematic chart. Choice of relaxation scheme for
different grid points.
Here,ξlimis a switch parameter to be defined. From practical tests it was found that
an efficient solver could be obtained usingξlim ≈ 0.3 and some underrelaxation in
both the processes. TypicallyωJacobi= 0.6 and ωG−S= 0.8.
A line relaxation can be described as follows. Let ˜pi, j denote the approximation to pi, j and ˜hi, j the associated approximation to hi, j. One relaxation sweep yields an improved approximation of ¯pi, j and ¯hi, j and consists of the following steps. The grid is scanned line by line, at each line applying changes to all the points of the line simultaneously, and, if distributive relaxation is used partly to the neighbouring lines. Subsequently, the new approximation ¯pi, j can be used to obtain ¯hi, j. For line relaxation in x-direction the changesδpi, j to be applied at given line j (and partly to the neighbouring lines) are solved from a system of equations of the form
∑
|k−i|<s
ak, jδpk, j = ri, j (3.10)
where ri, j is the vector of residuals and ak, j are the coefficients which exact defini-tion depends on the type of the relaxadefini-tion applied. For the described problem, the definition of ri, j and ak, j is given below. Theoretically, the matrix a is a full matrix. However, to obtain the line relaxation efficiency it is generally not needed to solve the system exactly. In practice it is often sufficient to take into account only the
terms in the summations related to the direct neighbours of a point (i, j). This is
justified because coefficient the a decrease with increasing distance from the point
(i, j). In particular it is sufficient to solve a three-diagonal system for Gauss-Seidel
line relaxation and hexadiagonal system for Jacobi distributive line relaxation. 3.3.1 Gauss-Seidel Line Relaxation
The discrete Reynolds Equation at grid point (i, j) is given by equation (3.6). We sweep line by line in the OX-direction. The residual ri, j at the grid point (i, j)
(a) (b)
Figure 3.3 Gauss-Seidel Line relaxation (a) and Jacobi distributive line relaxation
(b). Schematic charts.
depends on the given approximation pi, j, pi−1, j, pi+1, j, expression (3.11):
ri, j(pi−1, j, pi, j, pi+1, j) ≡ 1.5ρi, j(hi, j+ ∆hi, j) − 2ρi−1, j(hi−1, j+ ∆hi−1, j) + 0.5ρi−2, j(hi−2, j+ ∆hi−2, j) hx − (ei, j+ ei−1, j)pi−1, j− (ei−1, j+ 2ei, j+ ei+1, j)pi, j+ (ei, j+ ei+1, j)pi+1, j 2h2 x − (ei, j+ ei, j−1)pi, j−1− (ei, j−1+ 2ei, j+ ei, j+1)pi, j+ (ei, j+ ei, j+1)pi, j+1 2h2 y (3.11) where∆hi, j=∑l,kDik jlδplk;∆hi−1, j=∑l,kDi−1k jlδplk and∆hi−2 j=∑l,kDi−2k jlδplk. The corrections of pressure δpi, j (note that δpi, j = 0 at the first iteration) are solved from the linear three-diagonal system
ri, j= ai−1, jδpi−1, j+ ai, jδpi, j+ ai+1, jδpi+1, j for each fixed j (3.12) where coefficients ai, j = ∂p∂rii, j, j ˜ pi, j are pre-computed ai−1, j≡ ∂r i, j ∂pi−1, j = (ei, j+ ei−1, j) 2h2 x −1.5ρi, jD10− 2ρi−1, jD00+ 0.5ρi−2, jD10 hx ai, j≡ −∂ri, j ∂pi, j = (ei−1 j+ ei, j+ ei+1 j) 2h2 x −1.5ρi, jD10− 2ρi−1, jD00+ 0.5ρi−2, jD20 hx − (ei j−1+ 2ei, j+ ei+1 j) 2h2y ai+1, j≡ ∂ri, j ∂pi+1, j = (ei, j+ ei+1, j) 2h2 x −1.5ρi, jD10− 2ρi−1, jD20+ 0.5ρi−2, jD30 hx (3.13) where Diij j = D|i−i|,| j− j|≡ Dkl, and | i − i |= k and | j − j|= l. After all the pressure correctionsδpi, jare calculated, a new approximation ˜pi, jin each grid point is computed according to
¯
The film thickness updates∆hi, j are then calculated according to expressions (3.11) and applied in calculation of residuals at next relaxation sweep. A schematic chart of the Gauss-Seidel line relaxation is shown in Figure3.3(a).
3.4
Jacobi Distributive Line Relaxation
A second order distributive Jacobi relaxation was applied here. This technique is
described by many authors (see, for example, Brandt [2] or Venner [14]). A grid
point (i, j) is defined as a Jacobi distributive line relaxation point if ξ/(hxhy)
ξlim. For such a point the residual at the grid point (i, j) depends on the given
approximation on pi, j, pi−1, j, pi+1, j ri, j(pi−1, j, pi, j, pi+1, j) ≡ 1.5ρi, jhi, j− 2ρi−1, jhi−1, j+ 0.5ρi−2, jhi−2, j) hx − (ei, j+ ei−1, j)pi−1, j− (ei−1, j+ 2ei, j+ ei+1, j)pi, j+ (ei, j+ ei+1, j)pi+1, j 2h2 x − (ei, j+ ei, j−1)pi, j−1− (ei, j−1+ 2ei, j+ ei, j+1)pi, j+ (ei, j+ ei, j+1)pi, j+1 2h2 y (3.15)
The corrections of pressureδpi, j are solved from the linear hexadiagonal system
ri, j = ai−3, jδpi−3, j+ ai−2, jδpi−2, j+ ai−1, jδpi−1, j
+ ai, jδpi, j+ ai+1, jδpi+1, j+ ai+2, jδpi+2, j for each fixed j (3.16) where ai, j= ∂ri, j/∂pi, j−1/4 ∂ri, j/∂pi, j+∂ri, j/∂pi+1 j+∂ri, j/∂pi j−1+∂ri, j/∂pi j+1 for 1< i < niand 1< j < nj.
Let us introduce∆Di, j= Di, j−14 Di−1 j+ Di+1 j+ Di j−1+ Di1 j+1 ai−3, j= − 1.5ρi, j∆D30− 2ρi−1, j∆D20+ 0.5ρi−2, j∆D10 hx ai−2, j= − 1 4 (ei, j+ ei−1, j) 2h2 x −1.5ρi, j∆D20− 2ρi−1, j∆D10+ 0.5ρi−2, j∆D00 hx ai−1, j= (ei, j+ ei−1, j) 2h2 x +1 4 (ei−1, j+ 2ei, j+ ei+1, j) 2h2 x +1 4 (ei, j−1+ 2ei, j+ ei, j+1) 2h2 y −1.5ρi, j∆D10− 2ρi−1, j∆D00+ 0.5ρi−2, j∆D10 hx ai, j= − 5 4 (ei−1, j+ 2ei, j+ ei+1, j) 2h2 x −5 4 (ei, j−1+ 2ei, j+ ei, j+1) 2h2 y −1.5ρi, j∆D00− 2ρi−1, j∆D10+ 0.5ρi−2, j∆D20 hx ai+1, j= (e i, j+ ei−1, j) 2h2 x +1 4 (ei−1, j+ 2ei, j+ ei+1, j) 2h2 x +1 4 (ei, j−1+ 2ei, j+ ei, j+1) 2h2 y −1.5ρi, j∆D10− 2ρi−1, j∆D20+ 0.5ρi−2, j∆D30 hx ai+2, j= − 1 4 (ei, j+ ei−1, j) 2h2x − 1.5ρi, j∆D20− 2ρi−1, j∆D30+ 0.5ρi−2, j∆D40 hx (3.17) where Diij j = D|i−i|,| j− j|≡ Dkl, and| i − i|= k and | j − j|= l.
The hexadiagonal system is solved using Gaussian elimination and subsequently the changesδpi, j are applied to the line j. The changes are applied according to the scheme 1 4δi, j −10 −14 −10 0 −1 0 .
As a result of the distributive changes the new approximation ¯pi, j to pi, j is given by
¯ pi, j= ˜pi, j+ ωJacobi δpi, j− 1 4 δpi−1, j+ δpi+1, j+ δpi, j−1+ δpi, j+1 . (3.18)
After all the pressure corrections δpi, j are calculated, the pressure updates ¯pi, j is calculated according to expression (3.18) and applied in calculation of film thick-ness at next relaxation sweep. A schematic chart of the Jacobi distributive line relaxation is shown in Figure3.3(b).
The solution of the EHL problem is subject to a special condition, the
cavi-tation condition p 0 in Ω. This condition should be taken into account when
constructing the system of equations to be solved in the relaxation. Also when applying the changes the cavitation condition must be taken into account. The pro-cedure to apply the changes solved for the line j is the following. If a point (i, j) is a Gauss-Seidel point the changeωG−Sδpi, j is applied. If the resulting pressure
¯
pi, j= pi, j+ωG−Sδpi, j is smaller than zero it is set to zero. If the point(i, j) is a Ja-cobi distributive point the change ¯pi, j= ˜pi, j+ωJacobi
δpi, j−14 δpi−1, j+ δpi+1, j+ δpi, j−1+ δpi, j+1
is applied to the point(i, j). If the resulting pressure is smaller than zero it is set to zero. The net change, i.e. corrected for the cavitation condition,
is then distributed to the neighbouring points according to the distribution. Finally, the change to the central point is always applied. However, the changes to the neigh-bouring points are only applied if the current pressure at these points is larger then
zero and also after these changes have been applied the cavitation condition p 0
is imposed in these points.
4
Elastic Models of a Car Tyre
A wheel having a pneumatic tyre is an elastic body which has a very complex struc-ture. In this study of the viscous aquaplaning problem, two different elastic material models of the car tyre were considered, with an increasing degree of complexity.
4.1
Solid Isotropic Car Tyre Model
The mathematical model used here is restricted by the following assumptions
• The car tyre is a homogeneous, isotropic and perfectly elastic body • A frictionless contact is assumed
Let us recall that in assumptions above, the total deflection w(x,y) of a car tyre
loaded against the road becomes
w(x,y) = 2 πE +∞ −∞ +∞ −∞ p(x,y)dxdy (x − x)2+ (y − y)2 (4.1)
where Eis the effective elastic modulus.
Approximating the pressure profile by a piecewise constant function on a uni-form grid with mesh size hx and hyin the region
(x,y) ∈ R2|x
k− hx/2 ≤ x ≤ xk+ hx/2 ∧ yl− hy/2 ≤ y ≤ yl+ hy/2
,
where pkl= p(xk,yl), the elastic deformation (4a) at grid point (i, j) can be written as w(xi,yj) = wi, j ≈ 2 πE nx
∑
k=1 ny∑
l=1 Di jklpkl (4.2)The influence coefficients Di jkl are given by
Di jkl = xk+hx/2 xk−hx/2 yl+hy/2 yl−hy/2 dxdy (xi− x)2+ (yj− y)2 . (4.3)
Analytically calculated coefficients Di jklare given by the expression found by Love [9]
Di jkl= xpln yp+ x2 p+ y2p ym+ x2 p+ y2m +ymln ym+ x2m+ y2m yp+ x2 p+ y2m + xmln ym+ x2 m+ y2m yp+ x2 m+ y2p +ypln xp+ x2 p+ y2p xm+ x2 m+ y2p (4.4) where xp= xi− xk+ hx/2, xm = xi− xk− hx/2, yp = yj− yl+ hy/2, ym = yj−
yl− hy/2. The solid isotropic elastic model has a number of nice properties for numerical calculations as its kernel is symmetric and potential.
4.2
Pneumatic Car Tyre Model
Clearly the complex structure of a tyre does not lend itself to the analytical treat-ment, which is possible for solid isotropic bodies, nevertheless simplified model has been proposed which do account for the main features of the observed behaviour.
The stiff tread and cross-section of a motor tyre with internal pressure, when pressed into contact with a rigid plane surface, result in a contact path which is a
roughly elliptical, see Figure 4.1. The load is transmitted from the ground to the
rim through the walls. When the ground reaction is applied to the tyre the tension in the walls is decreased with consequent increase in curvature, thereby exerting an effective upthrust on the hub.
When considering a car tyre as an inflated membrane the contact pressure dis-tribution would be uniform and equal to the inflation pressure, whereas a solid tread would concentrate the pressure in the centre. The effect of bending stiffness of the tread is to introduce pressure peaks at the ends of the contact and support from the
walls gives high pressure at the edges, see Figure 4.2. The relative importance of
these different effects depends upon the design of the tyre.
Like in the model of solid elastic wheel in contact with a rigid ground, the math-ematical model of a pneumatic tyre could be described by a number of influence coefficients Dik jl, which are in convolution with contact pressure would result in a total elastic deflection of the tyre surface from the undeformed shape. The major difficulty is that the described effects with varied bending stiffness etc. will intro-duce strong nonlinearity into such a model.
To be able to account for the important effects of the car tyre behaviour and yet use the advantages of the theory of linear elasticity, the following model was proposed in this research.
A steady state of a car tyre pressed into contact with solid foundation was
calcu-Figure 4.1 Path of a car tyre in contact with the plane rigid surface.
(a) a (b) b
Figure 4.2 Shape of the car tyre in contact with road (a) and corresponding
pres-sure distribution (b). The nodes in the direct contact with road are shown with red points. The pressure distribution is given in finer resolution.
(a) a (b) b
Figure 4.3 Illustration on Green function of the elastic model of the pre-loaded
pneumatic car tyre. A unit point load is applied to the node (i = 3, j = 3, shown in red) on the pre-loaded tyre shape shown up side down, (a). An elastic response of the car tyre surface to the point unit load, (b). Deflections of all grid nodes from the initial pre-loaded state are shown on a coarse grid.
lated using a commercial finite element modelling software ABAQUS. The external load on the wheel was chosen 2500 N, which corresponds to the wheel load of a small car. In this pre-loaded condition, a point unit load was applied to all nodes in the grid subsequently, measuring the deflections of the other node points in the grid, see Chapter2.4and the illustration in Figure4.3.
In such way deflections in all grid points due to point unit loads were deter-mined and a Green function similar to the one used in the isotropic tyre model (4.2) was completed. The major differences of this approach from the previous one as following:
• The deflections are calculated from initially pre-loaded state (already
de-formed).
• To calculate the deflections, the pressure difference between hydrodynamic
and static contact pressures should be used, instead of pure hydrodynamic pressures.
The kernel (Green function) of the pre-loaded pneumatic car tyre model is not symmetrical neither potential, which demands a large number of additional storage calculations each time the deflections have to be determined.
4.3
Additional Remarks on Car Tyre Models
The major differences of the two models are as follows:
• The elastic properties of the isotropic elastic model remain the same through
the whole contact region. Under the action of external load, the pressure distribution reaches its maximum in the centre of the contact.
• The elastic properties of the pneumatic tyre vary through the contact:
– It is relatively stiff outside the contact
– It is very stiff on the edges of the contact because of bending moment of the stiff tread. This results in high edge pressures, Figure4.2(b)
– It is relatively “soft” in the centre of the contact (dominated by the pneu-matic pressure)
These properties have a major impact on the Elasto-Hydrodynamics of the two mod-els:
• The contact pressure increases gradually in the isotropic elastic model, and is
very low at the inlet of the contact. This makes it easier for the fluid to enter the contact region and build a separating fluid film.
• The rapidly increased static edge pressures and “hard” edges of the contact
prevent the fluid from entraining the contact. The fluid flow in the centre of the contact is of very limited importance because of the very small amount of fluid is flowing through the contact
This behaviour was confirmed in a number of numerical results. At comparable effective stiffness and contact loads, the solid isotropic model is more likely to pro-duce water films at lower velocities. However, the velocities must be very high to build a viscous fluid water films in tyre-to-road contact. The results are presented and discussed in the next chapter.
5
Results
5.1
General remarks on the viscous EHD pressures and contact
shapes
Let us recall the basic assumptions concerning fluid film build-up in a contact be-tween the car tyre and the road. These are:
• Incompressible and isoviscous fluid
• The inertia effects in fluid flow are neglected.
The two main features of the elastohydrodynamic fluid flow are
1. The geometry of the surfaces (convergent gap) enables the formation of a thin fluid layer separating them. Because of the viscosity of the fluid, its molecules stick to the surfaces and get drawn into the loaded area by the movement of the surfaces. Just because of the viscosity, it takes a certain time for the fluid to get into the contact and also to be pushed out from the contact zone. But a certain (very small) volume of fluid may remain even in the high pressure zone, thus building a continuous fluid layer effectively separating the surfaces. 2. High external load on the bodies leads to the elastic deformation of one or
both surfaces.
The fluid film formation in such contacts is strongly affected by the lubricant be-haviour. As soon as the surfaces are set in motion, the fluid becomes drawn into the contact by the action of purely viscous forces. The hydrodynamic pressure in the fluid layer will appear in the inlet of the contact region and propagate further into the contact. If the hydrodynamic pressure in the fluid layer is high enough, it will cause deformations of the surfaces and even can carry the whole contact load. Redistributions of the contact pressures will occur.
When the surfaces are completely separated by the fluid film, even very little difference in their velocities is enough to get them sliding against each other. The force is needed to shear the fluid layer is far lower than the friction force between rubber surface and an asphalt road. This is a brief explanation of the aquaplaning phenomena. To find out whether the viscous hydrodynamic pressure is high enough to carry the contact load or not, and what fluid layer thickness is expected in the contact between the tyre and the road, is the subject of this investigation.
Consider first static dry contacts, Figures5.1(a-d) and Figures5.2(a-d). The tyre becomes flat just in the contact region when pressed against the flat surface. The pressures are strictly limited to this area. Certain edge effects can be observed in
pressures form Figure 5.1(b). Some irregularities in the pressure distribution are
contributed by somewhat rough geometric description of the tyre shape on coarse computational grid. Contour plots Figures5.1(d) and 5.2(d) indicate no variations of the film thickness in the contact region.
Let us now take a look on the two examples of the viscous flow for both of the elastic models of the car tyre. The effects of the elastohydrodynamic fluid flow are illustrated in Figure5.1(e-h) for the case of solid isotropic elastic model of car tyre and Figure5.2(e-h) for pneumatic car tyre.
The running conditions for these cases were an external load of 2500 N, and
velocity was specially chosen to be very high to illustrate the effects of the EHD fluid film build-up. A few of the features of these results are common for both models
• Non-symmetrical “smooth” pressure distribution with pronounced
hydrody-namic pressure build-up in the inlet of the contact.
• Fluid film layer reaches its minima on the edges of the contact (at the side
lobes or at the inlet). Therefore two characteristics of the fluid film are often used, a minimum and a central film thickness.
Certain differences between the two EHD results can be seen which are explained by the differences in the elastic models of the solid isotropic elastic body and the pneumatic car tyre, see Chapter4for more details.
5.2
Running conditions for the numerical calculations
The following parameters were used to perform numerical simulations of the vis-cous EHD for the elastic models of the car tyre.
• Viscosity of water at 20◦C: 1.05 × 10−3 Pa× s = 1.05 × 10−3kg/m × s
• Density of pure water 103kg/m3
• Rolling velocity of the tyre was chosen from the practical needs to 20, 40, 60
120 and 200 km/h
• Modulus of elasticity of silicone rubber (solid isotropic model) 2.05 MPa,
Poisson ratio 0.5
• Elastic properties of the pneumatic tyre were modelled according to the
spec-ifications on a real test tyre.
It can be noticed in advance that no separated fluid films were detected for nei-ther model within the range of chosen velocities. This fact is discussed in Chapter6. Instead of fully separating fluid films, a combination of direct contacts between the surfaces and separated by the fluid flow were discovered. To be able to treat the mix of direct elastic contact with hydrodynamically supported contact, a specially designed algorithm was employed. This allows the complicated contact problems to be efficiently computed.
(a) Tyre shape in contact with flat rigid surface.
(b) Contact pressures.
(c) Contact shape (magnified).
(d) Contour plot of contact shape.
(e) Tyre shape in contact with flat rigid surface.
(f) Contact pressures.
(g) Contact shape (magnified).
(h) Contour plot of contact shape.
Figure 5.1 Illustration on static (a-d), 0 m/s, and dynamic (e-h), 1.66 × 103 m/s
(a) Tyre shape in contact with flat rigid surface.
(b) Contact pressures.
(c) Contact shape (magnified).
(d) Contour plot of contact shape.
(e) Tyre shape in contact with flat rigid surface.
(f) Contact pressures.
(g) Contact shape (magnified).
(h) Contour plot of contact shape.
Figure 5.2 Illustration on static (a-d), 0 m/s, and elastohydrodynamic (e-h) at
1.66 × 103m/s contacts of a pneumatic car tyre with a flat solid surface.
